A simple proof that the Riesz projection is bounded on $L^p(\mathbb{T})$ for $1<p<\infty$

Abstract

Let $\mathbf{P}$ denote the Riesz projection on the unit circle $\mathbb{T}$ and suppose that $1<p<\infty$. We present a simple proof of the bound $\Vert \mathbf{P}f{\Vert }_p\le \max (p,q)\Vert f{\Vert }_p$, where $f$ is in $L^p(\mathbb{T})$ and $p^{-1}+q^{-1}=1$. Our proof is a variation of a classical argument due to M. Riesz demonstrating that the Hilbert transform is bounded on $L^p(\mathbb{T})$.